Transparent faces even with Limit Selection to Visible
Your 3D View far clipping distance is set to a very high value. The floating point representation of depth in Blenders Viewport is not layed out to handle these high values.

You can easily recreate the problem, by 


Opening a new Blender file.
Selecting the cube and going into edit mode.
Switching to orthographic view.
Setting the View > Clip > End to 100000.


In your file, the end clipping distance is set to 1km. However, since your Scene > Unit Scale is set to 0.001, this value translates to ~1000000 BUs.